For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free graphs. We then extend our result to hold even for $(sP_3+P_6)$-free graphs for every $s\geq 0$ and initiate a complexity classification of Matching Cut for $H$-free graphs.

翻译：对于连接的图形$G=( V, E) 美元, 匹配的 $M\ subseteq E$ 是匹配的 $G$ 如果 $G- M$ 断开, 匹配的 E$是匹配的 $G$ 。 已知对于整数美元, 相应的决定问题匹配 Cut 是多米时间, 如果 $d\leq 2 美元, 直径的图形最多为 $d\leq 2 美元, 而 如果 $d\ geq 3 美元, 直径的图形最多为 $P- 3 美元, 且 NP- 3 美元, 我们证明受约束半径的图形是相同的二分法 。 对于 $H $, 如果它不包含 $H$, 一个图表是免费的, 图表是 $H$。 由于我们的结果, 我们可以解决以多元时间匹配的 $P_ 6 免费的图形, 将 Feghali 最新结果扩大到每 $P_ 5 。 然后我们的结果甚至为 $( 3+ P_ 6) exfreegage) $ 0 和 exporting cutting colding ricing cutting cut riculation exglection for $ $.